Toric Kempf–Ness Sets

In the theory of algebraic group actions on affine varieties, the concept of a Kempf–Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. Using recent achievements of “toric topology”, we show that an appropriate notion of a Kempf–Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the Batyrev–Cox “geometric invariant theory” approach to toric varieties. We proceed by studying the cohomology of these “toric” Kempf–Ness sets. In the case of projective nonsingular toric varieties the Kempf–Ness sets can be described as complete intersections of real quadrics in a complex space.